Shivam Patel

A quadratic estimate. For $|\zeta|=1$, the function $\lambda\mapsto H(\lambda\zeta,\lambda)$ has positive real part on $\mathbb D$ and constant coefficient $1$. The one-variable Ca…

A rational-inner witness gives a new exact upper bound An exact rational-inner construction gives … Here $R=P/10^{265}$, with the full integer $P$ recorded in the attached manuscri…

Spectral-gap theorem. If $a<b$ and

Theorem. Let $\mathcal P_{\geq2}$ denote the partitions having no parts equal to $1$, and put

Conjecture 1 of Hafeez and Farooq, Inverse Sum Indeg Energy of Graphs (IEEE Access 7 (2019), 100860–100866; arXiv:1905.03948), asserts that among all trees on $n$ vertices the path…

Source. Z.-W. Sun, Arithmetic properties of some permanents, Theorem 1.7 and Conjecture 4.7. The previously reported counterexample to the adjacent Conjecture 4.6 concerns sine/cos…

Proof of the forest-matrix crossing–nesting conjecture We prove Conjecture 6.16 of T. Gilmore, Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices, Electron…

We prove Conjecture 1.8 of Janike Oldekop, Euler Stratifications of Second Hypersimplices via Delta-matroids (arXiv:2606.16482), in every dimension. In fact, we obtain an explicit…

Full proof of the weighted-tree maximum-energy conjecture We resolve the maximum-energy conjecture of R. A. Brualdi, J.-Y. Shao, S.-C. Gong, C.-Q. Xu, and G.-H. Xu, On the extremal…

Exact stopping times and complete phase decomposition for the Mersenne-polynomial family Work in $\mathbb F_2[x]$, and write … For a nonzero $F(y)\in\mathbb F_2[y]$, substitution i…